Year 2013

Thursday 28th March 2013 at 14h
Pascal Koiran
(Ecole Normale Supérieure de Lyon),
A Wronskian approach to the real tau-conjecture

Abstract:(Hide abstracts)
According to Shub and Smale's tau-conjecture, the number of integer roots of a univariate polynomial should be polynomially bounded in the size of the smallest (constant free) straight-line program computing it. This statement becomes provably false if one counts real roots instead of integer roots. I have proposed a real version of the tau-conjecture where the attention is restricted to straight-line programs of a special form: the sums of products of sparse polynomials. This conjecture implies that the permanent polynomial cannot be computed by polynomial-size arithmetic circuits. The complexity of the permanent in the arithmetic circuit model is a long standing open problem, which can be thought of as an algebraic version of P versus NP. In this talk I will present the real tau-conjecture and its consequence for the permanent. If time allows, I will introduce a new tool in this context: the Wronksian determinant. This leads to some modest progress on the real tau-conjecture, and to new bounds on the number of solutions of sparse systems of polynomial equations. The latter bounds seem to be of independent interest from the point of view of real algebraic geometry.

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In this talk I will illustrate how to use basic results in Ehrhart theory to solve a problem on discriminants. I will introduce all the necessary notions such as Ehrhart polynomials and lattice polytopes. As it turns out, the problem will be reduced to a question about binomial coefficients.